Finalizing tentative matches from truncated preference lists

Consider the standard hospitals/residents problem, or the two-sided many-to-one stable matching problem, and assume that the true preference lists of both sides are complete (containing all members of the opposite side) and strict (having no ties). The lists actually submitted, however, are truncated. Let I be such a truncated instance. When we apply the resident-proposing deferred acceptance algorithm of Gale and Shapley to I, the algorithm produces a set of tentative matches (resident-hospital pairs). We say that a tentative match in this set is finalizable in I if it is in the resident-optimal stable matching for every completion of I (a complete instance of which I is a truncation). We study the problem we call FTM (Finalizability of Tentative Matches) of deciding if a given tentative match is finalizable in a given truncated instance. We first show that FTM is coNP-complete, even in the stable marriage case where the quota of each hospital is restricted to be 1. We then introduce and study a special case: we say that a truncated instance is resident-minimal, if further truncation of the preference lists of the residents inevitably changes the set of tentative matches. Residentminimal instances are not only practically motivated but also useful in computations for the general case. We give a computationally useful characterization of negative instances of FTM in this special case, which, for instance, can be used to formulate an integer program for FTM. For the stable marriage case, in particular, this characterization yields a polynomial time algorithm to solve FTM for resident-minimal instances. On the other hand, we show that FTM remains coNP-complete for residentminimal instances, if the maximum quota of the hospitals is 2 or larger. We also give a polynomial-time decidable sufficient condition for a tentative match to be finalizable in the general case. Simulations show that this sufficient condition is extremely useful in a two-round matching procedure based on FTM for a certain type of student-supervisor markets.